library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(dplyr)
library(fGarch)

## Loading required package: timeDate

## Loading required package: timeSeries

## Loading required package: fBasics

library(dynlm)

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following object is masked from 'package:timeSeries':
## 
##     time<-

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

library(tseries)

## Registered S3 method overwritten by 'xts':
##   method     from
##   as.zoo.xts zoo

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

library(astsa)

## 
## Attaching package: 'astsa'

## The following object is masked from 'package:fBasics':
## 
##     nyse

library(xts)

## 
## Attaching package: 'xts'

## The following objects are masked from 'package:dplyr':
## 
##     first, last

library(fpp)

## Loading required package: forecast

## 
## Attaching package: 'forecast'

## The following object is masked from 'package:astsa':
## 
##     gas

## Loading required package: fma

## 
## Attaching package: 'fma'

## The following objects are masked from 'package:astsa':
## 
##     chicken, sales

## Loading required package: expsmooth

## Loading required package: lmtest

## 
## Attaching package: 'fpp'

## The following object is masked from 'package:astsa':
## 
##     oil

library(TSA)

## Registered S3 methods overwritten by 'TSA':
##   method       from    
##   fitted.Arima forecast
##   plot.Arima   forecast

## 
## Attaching package: 'TSA'

## The following objects are masked from 'package:timeDate':
## 
##     kurtosis, skewness

## The following objects are masked from 'package:stats':
## 
##     acf, arima

## The following object is masked from 'package:utils':
## 
##     tar

library(forecast)
library(dplR)
library(tseries)

Research Indicator: Employment, Persons, Number of

Indicator Code: LE_PE_NUM

# read file 
ifs <- read.csv("IFS_04-02-2020 18-49-44-40.csv",stringsAsFactors =FALSE)
colnames(ifs)=c("Country_Name", "Country_Code", "Indicator_Name", "Indicator_Code", "Time_Period", "Value", "Status")
# remove Status column 
ifs=ifs[, c(1:6)]
# dataset preview
head(ifs)

##    Country_Name Country_Code
## 1 United States          111
## 2 United States          111
## 3 United States          111
## 4 United States          111
## 5 United States          111
## 6 United States          111
##                                                Indicator_Name
## 1 International Reserves, Official Reserve Assets, US Dollars
## 2 International Reserves, Official Reserve Assets, US Dollars
## 3 International Reserves, Official Reserve Assets, US Dollars
## 4 International Reserves, Official Reserve Assets, US Dollars
## 5 International Reserves, Official Reserve Assets, US Dollars
## 6 International Reserves, Official Reserve Assets, US Dollars
##   Indicator_Code Time_Period       Value
## 1       RAFA_USD      1980M5 21917557876
## 2       RAFA_USD      1980M9 22994402034
## 3       RAFA_USD     1980M11 25673025848
## 4       RAFA_USD      1981M4 29692779278
## 5       RAFA_USD      1981M9 29715202935
## 6       RAFA_USD      1982M4 31555856317

print(colnames(ifs))

## [1] "Country_Name"   "Country_Code"   "Indicator_Name" "Indicator_Code"
## [5] "Time_Period"    "Value"

United States

## Select country
df1_us=subset(ifs, (Indicator_Code=='LE_PE_NUM') & (Country_Name=='United States'))

## Reaarange time
df1_us$Time_Period <- gsub("M", "/", df1_us$Time_Period)

df1_us['date'] = '/01'
df1_us$Time_Period <- paste(df1_us$Time_Period, df1_us$date,  sep="")

df1_us$Time_Period = as.Date(df1_us$Time_Period, "%Y/%m/%d")

df1_us = df1_us[order(df1_us$Time_Period),]
df1_us = subset(df1_us, select = -c(date))
rownames(df1_us)=NULL

df11=ts(df1_us$Value, frequency = 12, start = c(1980, 1))
plot(df11, main="United States: Employment, Persons, Number of")

## 1.Decompose Analysis

# Decompose
plot(decompose(df11))

## There is a obvious growing trend with little drop in 2008-2010 period in the data
## There is seasonality in the data

2.Spectral Analysis

sp=spectrum(df11, log='no')

sp$fre[which.max(sp$spec)] #0.025

## [1] 0.025

1/sp$fre[which.max(sp$spec)] # 40 month

## [1] 40

max(sp$spec) #estimate of spectral density at predominant period to be 1.344801e+14

## [1] 1.344801e+14

U = qchisq(.025,2)
L = qchisq(.975,2)
2*max(sp$spec)/L #lower bound for spectral density

## [1] 3.645555e+13

2*max(sp$spec)/U #upper bound for spectral density

## [1] 5.311681e+15

# The 95% confidence interval is [3.645555e+13, 5.311681e+15]

3.ARIMA model

df11.diff=diff(df11)#first order difference,d=1
plot(df11.diff, main=" difflog data")

acf(df11.diff,lag.max = 100) ## q = 1

pacf(df11.diff) #p=1,2

arima111 = Arima(df11,order = c(1,1,1),include.drift = TRUE)
arima211  =Arima(df11,order = c(2,1,1),include.drift = TRUE)
AIC(arima111)

## [1] 14295.02

AIC(arima211)

## [1] 14274.71

BIC(arima111)

## [1] 14311.69

BIC(arima211)

## [1] 14295.55

auto.arima(df11,seasonal = F)

## Series: df11 
## ARIMA(2,1,2) with drift 
## 
## Coefficients:
##           ar1      ar2     ma1     ma2      drift
##       -0.0091  -0.9785  0.1329  0.9694  125233.35
## s.e.   0.0107   0.0099  0.0135  0.0152   33063.04
## 
## sigma^2 estimated as 4.713e+11:  log likelihood=-7086.8
## AIC=14185.61   AICc=14185.79   BIC=14210.61

AIC(auto.arima(df11,seasonal = F))

## [1] 14185.61

BIC(auto.arima(df11,seasonal = F))

## [1] 14210.61

#Seasonal ARIMA
fit1 <- auto.arima(df11,trace = T) # Best model: ARIMA(3,0,1)(0,1,2)[12] with drift

## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,0,2)(1,1,1)[12] with drift         : 12979.15
##  ARIMA(0,0,0)(0,1,0)[12] with drift         : 14381.27
##  ARIMA(1,0,0)(1,1,0)[12] with drift         : 13067.65
##  ARIMA(0,0,1)(0,1,1)[12] with drift         : 13870.24
##  ARIMA(0,0,0)(0,1,0)[12]                    : 14628.2
##  ARIMA(2,0,2)(0,1,1)[12] with drift         : 12968.43
##  ARIMA(2,0,2)(0,1,0)[12] with drift         : 13142.5
##  ARIMA(2,0,2)(0,1,2)[12] with drift         : 12969.52
##  ARIMA(2,0,2)(1,1,0)[12] with drift         : 13067
##  ARIMA(2,0,2)(1,1,2)[12] with drift         : 12969.35
##  ARIMA(1,0,2)(0,1,1)[12] with drift         : 12985.5
##  ARIMA(2,0,1)(0,1,1)[12] with drift         : 12995.95
##  ARIMA(3,0,2)(0,1,1)[12] with drift         : 12966.81
##  ARIMA(3,0,2)(0,1,0)[12] with drift         : 13132.82
##  ARIMA(3,0,2)(1,1,1)[12] with drift         : 12989.08
##  ARIMA(3,0,2)(0,1,2)[12] with drift         : 12966.32
##  ARIMA(3,0,2)(1,1,2)[12] with drift         : 12973.2
##  ARIMA(3,0,1)(0,1,2)[12] with drift         : 12965.76
##  ARIMA(3,0,1)(0,1,1)[12] with drift         : 12965.83
##  ARIMA(3,0,1)(1,1,2)[12] with drift         : 12970.98
##  ARIMA(3,0,1)(1,1,1)[12] with drift         : 12987.04
##  ARIMA(2,0,1)(0,1,2)[12] with drift         : 12993.09
##  ARIMA(3,0,0)(0,1,2)[12] with drift         : 12980.94
##  ARIMA(4,0,1)(0,1,2)[12] with drift         : 12968.13
##  ARIMA(2,0,0)(0,1,2)[12] with drift         : 12989.24
##  ARIMA(4,0,0)(0,1,2)[12] with drift         : 12972.65
##  ARIMA(4,0,2)(0,1,2)[12] with drift         : 12969.77
##  ARIMA(3,0,1)(0,1,2)[12]                    : Inf
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(3,0,1)(0,1,2)[12] with drift         : 13322.43
## 
##  Best model: ARIMA(3,0,1)(0,1,2)[12] with drift

AIC(fit1)

## [1] 13322.11

sarima(df11,3,0,1,0,1,2,12)

## initial  value 14.386342 
## iter   2 value 13.562982
## iter   3 value 13.502828
## iter   4 value 13.324031
## iter   5 value 13.296946
## iter   6 value 13.014887
## iter   7 value 12.920372
## iter   8 value 12.898252
## iter   9 value 12.898007
## iter  10 value 12.888989
## iter  11 value 12.888153
## iter  12 value 12.887149
## iter  13 value 12.886341
## iter  14 value 12.884883
## iter  15 value 12.881125
## iter  16 value 12.869233
## iter  17 value 12.866209
## iter  18 value 12.865978
## iter  19 value 12.857886
## iter  20 value 12.857396
## iter  21 value 12.856502
## iter  22 value 12.856148
## iter  23 value 12.854197
## iter  24 value 12.853615
## iter  25 value 12.853177
## iter  26 value 12.853088
## iter  27 value 12.853067
## iter  28 value 12.853035
## iter  29 value 12.852932
## iter  30 value 12.852819
## iter  31 value 12.852704
## iter  32 value 12.852627
## iter  33 value 12.852626
## iter  34 value 12.852623
## iter  35 value 12.852621
## iter  36 value 12.852618
## iter  37 value 12.852615
## iter  38 value 12.852614
## iter  39 value 12.852613
## iter  40 value 12.852612
## iter  41 value 12.852612
## iter  41 value 12.852612
## iter  41 value 12.852612
## final  value 12.852612 
## converged
## initial  value 12.862009 
## iter   2 value 12.861765
## iter   3 value 12.861551
## iter   4 value 12.861413
## iter   5 value 12.860991
## iter   6 value 12.860384
## iter   7 value 12.859646
## iter   8 value 12.859216
## iter   9 value 12.858946
## iter  10 value 12.858638
## iter  11 value 12.858282
## iter  12 value 12.858117
## iter  13 value 12.858050
## iter  14 value 12.858042
## iter  15 value 12.858038
## iter  16 value 12.858034
## iter  17 value 12.858032
## iter  18 value 12.858027
## iter  19 value 12.858024
## iter  20 value 12.858015
## iter  21 value 12.858013
## iter  22 value 12.858009
## iter  23 value 12.858009
## iter  23 value 12.858008
## iter  23 value 12.858008
## final  value 12.858008 
## converged

## $fit
## 
## Call:
## stats::arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, 
##     Q), period = S), xreg = constant, transform.pars = trans, fixed = fixed, 
##     optim.control = list(trace = trc, REPORT = 1, reltol = tol))
## 
## Coefficients:
##          ar1      ar2      ar3      ma1     sma1     sma2   constant
##       1.8449  -0.7614  -0.0864  -0.8606  -0.7253  -0.0351  119339.19
## s.e.  0.0763   0.1237   0.0523   0.0617   0.0521   0.0524   17725.74
## 
## sigma^2 estimated as 1.434e+11:  log likelihood = -6653.06,  aic = 13322.11
## 
## $degrees_of_freedom
## [1] 459
## 
## $ttable
##             Estimate         SE  t.value p.value
## ar1           1.8449     0.0763  24.1908  0.0000
## ar2          -0.7614     0.1237  -6.1545  0.0000
## ar3          -0.0864     0.0523  -1.6512  0.0994
## ma1          -0.8606     0.0617 -13.9417  0.0000
## sma1         -0.7253     0.0521 -13.9341  0.0000
## sma2         -0.0351     0.0524  -0.6696  0.5034
## constant 119339.1945 17725.7407   6.7325  0.0000
## 
## $AIC
## [1] 27.92896
## 
## $AICc
## [1] 27.92946
## 
## $BIC
## [1] 27.99847

# The model is a good fit according to the disgnose plots.

Prediction

pred1=forecast(fit1,h=48)
pred1 # The last 2 columns are 95% prediction intervals lower bound and upper bound

##          Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## Nov 2019      158514903 158025460 159004347 157766364 159263443
## Dec 2019      158023944 157337176 158710713 156973622 159074266
## Jan 2020      156850068 155990978 157709158 155536203 158163933
## Feb 2020      157824566 156808208 158840924 156270181 159378951
## Mar 2020      158367594 157203612 159531577 156587437 160147752
## Apr 2020      158860109 157555781 160164437 156865312 160854907
## May 2020      159227543 157788910 160666176 157027344 161427742
## Jun 2020      159634382 158066804 161201960 157236978 162031786
## Jul 2020      159924387 158232835 161615939 157337382 162511393
## Aug 2020      159456801 157646020 161267582 156687451 162226152
## Sep 2020      159763796 157838395 161689196 156819150 162708441
## Oct 2020      160076479 158040986 162111972 156963461 163189497
## Nov 2020      160028517 157841956 162215078 156684460 163372574
## Dec 2020      159505235 157177486 161832984 155945250 163065220
## Jan 2021      158315168 155851260 160779076 154546945 162083390
## Feb 2021      159248687 156654019 161843356 155280484 163216891
## Mar 2021      159774908 157054811 162495005 155614878 163934937
## Apr 2021      160253584 157413366 163093802 155909846 164597322
## May 2021      160608140 157653078 163563202 156088762 165127517
## Jun 2021      160998838 157934168 164063507 156311830 165685846
## Jul 2021      161267651 158098561 164436740 156420947 166114355
## Aug 2021      160727268 157458886 163995650 155728709 165725827
## Sep 2021      161012291 157649672 164374910 155869609 166154974
## Oct 2021      161312380 157860494 164764265 156033177 166591582
## Nov 2021      161246675 157683095 164810254 155796650 166696700
## Dec 2021      160707885 157039634 164376137 155097779 166317992
## Jan 2022      159503806 155735237 163272375 153740277 165267335
## Feb 2022      160424816 156560570 164289062 154514962 166334671
## Mar 2022      160939966 156984674 164895257 154890870 166989062
## Apr 2022      161408954 157367238 165450670 155227682 167590225
## May 2022      161755145 157631589 165878702 155448710 168061581
## Jun 2022      162138746 157937875 166339617 155714068 168563423
## Jul 2022      162401670 158127936 166675404 155865557 168937782
## Aug 2022      161856550 157514312 166198787 155215671 168497428
## Sep 2022      162137930 157731445 166544415 155398792 168877068
## Oct 2022      162435413 157968819 166902008 155604346 169266481
## Nov 2022      162368086 157825503 166910669 155420805 169315367
## Dec 2022      161828601 157215294 166441908 154773157 168884045
## Jan 2023      160624699 155943987 165305411 153466168 167783230
## Feb 2023      161546706 156802100 166291312 154290457 168802955
## Mar 2023      162063620 157258606 166868633 154714985 169412254
## Apr 2023      162535088 157673125 167397051 155099357 169970818
## May 2023      162884426 157968926 167799927 155366817 170402036
## Jun 2023      163271791 158306100 168237482 155677422 170866160
## Jul 2023      163539049 158526438 168551660 155872922 171205176
## Aug 2023      162998788 157942438 168055138 155265768 170731808
## Sep 2023      163285506 158188500 168382513 155490307 171080705
## Oct 2023      163588764 158454075 168723453 155735935 171441593

autoplot(pred1) #plot with confidence interval

4.ARCH/GARCH model

# fit ARIMA model without seasonal term 
fit2=auto.arima(df11,  seasonal = FALSE, trace = TRUE) # ARIMA(2,1,2)

## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)            with drift         : 14160.76
##  ARIMA(0,1,0)            with drift         : 14289.43
##  ARIMA(1,1,0)            with drift         : 14269.58
##  ARIMA(0,1,1)            with drift         : 14265.89
##  ARIMA(0,1,0)                               : 14299.42
##  ARIMA(1,1,2)            with drift         : 14257.35
##  ARIMA(2,1,1)            with drift         : 14249.3
##  ARIMA(3,1,2)            with drift         : Inf
##  ARIMA(2,1,3)            with drift         : 14217.65
##  ARIMA(1,1,1)            with drift         : 14268.86
##  ARIMA(1,1,3)            with drift         : 14214.89
##  ARIMA(3,1,1)            with drift         : Inf
##  ARIMA(3,1,3)            with drift         : 14164.02
##  ARIMA(2,1,2)                               : 14170.95
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(2,1,2)            with drift         : 14185.79
## 
##  Best model: ARIMA(2,1,2)            with drift

#take the log transformation and first order difference
df11.difflog=diff(log(df11))
plot(df11.difflog, main=" difflog data")

acf(df11.difflog, lag.max = 20) # from pacf plot, we could observe the seasonality

pacf(df11.difflog, lag.max = 20)# from pacf plot, AR(1)

acf2(df11^2) # accoring to pacf plot, ARCH(1) model

##        ACF  PACF
##  [1,] 0.99  0.99
##  [2,] 0.98 -0.03
##  [3,] 0.98  0.01
##  [4,] 0.97  0.02
##  [5,] 0.96 -0.01
##  [6,] 0.95  0.03
##  [7,] 0.95  0.06
##  [8,] 0.94  0.01
##  [9,] 0.93 -0.02
## [10,] 0.93  0.05
## [11,] 0.92 -0.03
## [12,] 0.91 -0.04
## [13,] 0.91 -0.12
## [14,] 0.90 -0.02
## [15,] 0.89  0.02
## [16,] 0.88  0.00
## [17,] 0.87  0.00
## [18,] 0.86  0.02
## [19,] 0.86  0.06
## [20,] 0.85  0.01
## [21,] 0.84 -0.02
## [22,] 0.84  0.05
## [23,] 0.83 -0.01
## [24,] 0.82 -0.04
## [25,] 0.82 -0.10
## [26,] 0.81 -0.03
## [27,] 0.80  0.02
## [28,] 0.79  0.01
## [29,] 0.78  0.00
## [30,] 0.77  0.01
## [31,] 0.76  0.03
## [32,] 0.76  0.02
## [33,] 0.75  0.00
## [34,] 0.75  0.03
## [35,] 0.74 -0.01
## [36,] 0.73 -0.02
## [37,] 0.72 -0.09
## [38,] 0.71 -0.02
## [39,] 0.71  0.00
## [40,] 0.70  0.02
## [41,] 0.69 -0.01
## [42,] 0.68  0.02
## [43,] 0.67  0.04
## [44,] 0.67  0.01
## [45,] 0.66  0.00
## [46,] 0.66  0.03
## [47,] 0.65  0.00
## [48,] 0.64  0.00

# fit ARIMA(2,1,2) model, then plot the residuals ACF and PACF
arima212=arima(df11, order=c(2,1,2))
residual_arima212=arima212$residuals
acf2(residual_arima212)

##         ACF  PACF
##  [1,]  0.17  0.17
##  [2,]  0.22  0.20
##  [3,] -0.14 -0.22
##  [4,] -0.16 -0.16
##  [5,] -0.23 -0.11
##  [6,] -0.24 -0.18
##  [7,] -0.16 -0.10
##  [8,] -0.19 -0.18
##  [9,] -0.21 -0.31
## [10,]  0.24  0.29
## [11,]  0.26  0.23
## [12,]  0.70  0.59
## [13,]  0.17  0.04
## [14,]  0.22 -0.01
## [15,] -0.17 -0.14
## [16,] -0.20 -0.08
## [17,] -0.25 -0.08
## [18,] -0.20  0.00
## [19,] -0.21 -0.04
## [20,] -0.21 -0.01
## [21,] -0.22 -0.15
## [22,]  0.22 -0.04
## [23,]  0.23 -0.06
## [24,]  0.63  0.20
## [25,]  0.18  0.02
## [26,]  0.22  0.08
## [27,] -0.22 -0.11
## [28,] -0.18  0.00
## [29,] -0.23  0.01
## [30,] -0.21 -0.04
## [31,] -0.22  0.00
## [32,] -0.21  0.02
## [33,] -0.22 -0.08
## [34,]  0.22  0.03
## [35,]  0.21 -0.06
## [36,]  0.61  0.13
## [37,]  0.14 -0.09
## [38,]  0.20 -0.02
## [39,] -0.22 -0.07
## [40,] -0.20 -0.05
## [41,] -0.21  0.04
## [42,] -0.20  0.01
## [43,] -0.22 -0.01
## [44,] -0.22 -0.04
## [45,] -0.20 -0.01
## [46,]  0.24  0.07
## [47,]  0.19 -0.02
## [48,]  0.59  0.05

acf2(residual_arima212^2)

##         ACF  PACF
##  [1,]  0.02  0.02
##  [2,] -0.07 -0.07
##  [3,] -0.08 -0.08
##  [4,]  0.01  0.01
##  [5,] -0.01 -0.02
##  [6,] -0.09 -0.10
##  [7,] -0.01 -0.01
##  [8,]  0.04  0.02
##  [9,] -0.07 -0.09
## [10,] -0.12 -0.12
## [11,]  0.11  0.10
## [12,]  0.39  0.37
## [13,]  0.06  0.06
## [14,] -0.11 -0.07
## [15,] -0.06 -0.02
## [16,]  0.02  0.02
## [17,] -0.01  0.00
## [18,] -0.09 -0.04
## [19,] -0.01 -0.02
## [20,]  0.01 -0.03
## [21,] -0.07 -0.03
## [22,] -0.10 -0.02
## [23,]  0.03 -0.06
## [24,]  0.38  0.24
## [25,]  0.05  0.04
## [26,] -0.08 -0.01
## [27,] -0.09 -0.05
## [28,]  0.00 -0.03
## [29,]  0.01  0.03
## [30,] -0.08 -0.01
## [31,]  0.00  0.01
## [32,]  0.02 -0.01
## [33,] -0.08 -0.04
## [34,] -0.09  0.01
## [35,]  0.03 -0.03
## [36,]  0.42  0.24
## [37,]  0.00 -0.06
## [38,] -0.11 -0.05
## [39,] -0.08 -0.01
## [40,]  0.04  0.04
## [41,] -0.06 -0.09
## [42,] -0.05  0.02
## [43,]  0.00  0.02
## [44,]  0.03  0.00
## [45,] -0.05  0.02
## [46,] -0.04  0.08
## [47,]  0.08  0.02
## [48,]  0.32  0.07

garch_us1=garch(residual_arima212,order=c(3,3),trace=F)

## Warning in garch(residual_arima212, order = c(3, 3), trace = F): singular
## information

summary(garch_us1)

## 
## Call:
## garch(x = residual_arima212, order = c(3, 3), trace = F)
## 
## Model:
## GARCH(3,3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.3137 -0.4171  0.1818  0.7922  2.9313 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 3.263e+11          NA       NA       NA
## a1 5.171e-02          NA       NA       NA
## a2 4.897e-03          NA       NA       NA
## a3 3.661e-03          NA       NA       NA
## b1 7.308e-02          NA       NA       NA
## b2 7.493e-02          NA       NA       NA
## b3 7.711e-02          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 4.6405, df = 2, p-value = 0.09825
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.33624, df = 1, p-value = 0.562

AIC(garch_us1)

## [1] 14142.81

garch_us2=garch(residual_arima212,order=c(0,3),trace=F) # (q,p)

## Warning in garch(residual_arima212, order = c(0, 3), trace = F): singular
## information

summary(garch_us2)

## 
## Call:
## garch(x = residual_arima212, order = c(0, 3), trace = F)
## 
## Model:
## GARCH(0,3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4527 -0.4341  0.1904  0.8166  3.0020 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 3.962e+11          NA       NA       NA
## a1 6.243e-02          NA       NA       NA
## a2 1.700e-03          NA       NA       NA
## a3 9.513e-04          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 4.9537, df = 2, p-value = 0.08401
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.71223, df = 1, p-value = 0.3987

AIC(garch_us2) # the AIC is smaller, so ARCH(3) model is better

## [1] 14139.16

United Kingdom

## Select country
df1_uk=subset(ifs, (Indicator_Code=='LE_PE_NUM') & (Country_Name=='United Kingdom'))

df1_uk$Time_Period <- gsub("M", "/", df1_uk$Time_Period)

df1_uk['date'] = '/01'
df1_uk$Time_Period <- paste(df1_uk$Time_Period, df1_uk$date,  sep="")

df1_uk$Time_Period = as.Date(df1_uk$Time_Period, "%Y/%m/%d")

df1_uk = df1_uk[order(df1_uk$Time_Period),]
df1_uk = subset(df1_uk, select = -c(date))
rownames(df1_uk)=NULL

df22=ts(df1_uk$Value, frequency = 12, start = c(1992, 4))
plot(df22, main="United Kingdom: Employment, Persons, Number of")

1. decompose Analysis

plot(decompose(df22))

## There is a obvious growing trend with little drop in 2008-2010 period in the data
## There is seasonality in the data

2. Spectral Analysis

sp=spectrum(df22, log='no')

# find the cycle with predominant frequecny 
sp$fre[which.max(sp$spec)] #  0.066

## [1] 0.06666667

# find the cycle of predominant frequecny 
1/sp$fre[which.max(sp$spec)] # 15 month

## [1] 15

max(sp$spec) #estimate of spectral density at predominant period which is close to 899257074378.

## [1] 899257074378

U = qchisq(.025,2)
L = qchisq(.975,2)

2*max(sp$spec)/L #lower bound for spectral density

## [1] 243775131599

2*max(sp$spec)/U #upper bound for spectral density

## [1] 3.551876e+13

# The 95% confidence interval is [243775131599, 3.551876e+13]

3. ARIMA model

df22.diff = diff(df22)
plot(df22.diff)

acf2(df22.diff)#p = 4,5 , q = 1,2

##         ACF  PACF
##  [1,]  0.45  0.45
##  [2,]  0.10 -0.13
##  [3,]  0.00  0.02
##  [4,]  0.05  0.07
##  [5,] -0.27 -0.42
##  [6,] -0.45 -0.20
##  [7,] -0.25  0.09
##  [8,]  0.04  0.12
##  [9,]  0.00 -0.07
## [10,]  0.09  0.19
## [11,]  0.30  0.14
## [12,]  0.48  0.17
## [13,]  0.28  0.04
## [14,]  0.00 -0.14
## [15,] -0.03  0.02
## [16,]  0.01  0.10
## [17,] -0.23 -0.13
## [18,] -0.41 -0.12
## [19,] -0.28 -0.02
## [20,]  0.01  0.05
## [21,] -0.04 -0.16
## [22,]  0.01  0.11
## [23,]  0.29  0.19
## [24,]  0.50  0.08
## [25,]  0.29  0.04
## [26,]  0.05  0.00
## [27,]  0.09  0.11
## [28,]  0.03 -0.07
## [29,] -0.26 -0.05
## [30,] -0.44 -0.10
## [31,] -0.26  0.02
## [32,] -0.05 -0.02
## [33,] -0.05 -0.07
## [34,]  0.00  0.03
## [35,]  0.28  0.06
## [36,]  0.43  0.01
## [37,]  0.27  0.03
## [38,]  0.04  0.01
## [39,]  0.06  0.03
## [40,]  0.01 -0.03
## [41,] -0.29 -0.09
## [42,] -0.46 -0.04
## [43,] -0.28 -0.02
## [44,] -0.05  0.03
## [45,] -0.04  0.00
## [46,] -0.02 -0.01
## [47,]  0.26  0.01
## [48,]  0.41  0.01

arima411 = Arima(df22,order = c(4,1,1),include.drift = T)
arima511 = Arima(df22,order = c(5,1,1),include.drift = T)
arima412 = Arima(df22,order = c(4,1,2),include.drift = T)
arima512 = Arima(df22,order = c(5,1,2),include.drift = T)
AIC(arima411)

## [1] 8035.022

AIC(arima511)

## [1] 7979.001

AIC(arima412)

## [1] 8012.329

AIC(arima512)

## [1] 7954.431

BIC(arima411)

## [1] 8061.509

BIC(arima511)

## [1] 8009.272

BIC(arima412)

## [1] 8042.599

BIC(arima512)

## [1] 7988.485

auto.arima(df22,seasonal = F)

## Series: df22 
## ARIMA(5,1,1) with drift 
## 
## Coefficients:
##          ar1      ar2      ar3     ar4      ar5      ma1      drift
##       0.7583  -0.2536  -0.0473  0.2995  -0.4508  -0.2670  22452.337
## s.e.  0.0835   0.0718   0.0652  0.0631   0.0500   0.0853   2975.167
## 
## sigma^2 estimated as 2.608e+09:  log likelihood=-3981.5
## AIC=7979   AICc=7979.46   BIC=8009.27

## arima512 has a lower AIC and BIC.

fit22 <- auto.arima(df22) #  Best model: ARIMA(2,1,3)(2,1,1)[12] 
fit22

## Series: df22 
## ARIMA(2,1,3)(2,1,1)[12] 
## 
## Coefficients:
##          ar1     ar2     ma1      ma2      ma3     sar1     sar2     sma1
##       0.0434  0.8824  0.2511  -0.7471  -0.2454  -0.1288  -0.0045  -0.8871
## s.e.  0.0791  0.0754  0.0932   0.0862   0.0553   0.0721   0.0681   0.0483
## 
## sigma^2 estimated as 1.981e+09:  log likelihood=-3798.04
## AIC=7614.08   AICc=7614.67   BIC=7647.79

sarima(df22,2,1,3,2,1,1,12) # not a very good fit

## initial  value 11.084866 
## iter   2 value 10.871398
## iter   3 value 10.773249
## iter   4 value 10.758966
## iter   5 value 10.739943
## iter   6 value 10.730917
## iter   7 value 10.727753
## iter   8 value 10.722921
## iter   9 value 10.721991
## iter  10 value 10.720711
## iter  11 value 10.716986
## iter  12 value 10.714496
## iter  13 value 10.714321
## iter  14 value 10.714122
## iter  15 value 10.714107
## iter  16 value 10.714095
## iter  17 value 10.714094
## iter  18 value 10.714093
## iter  18 value 10.714093
## final  value 10.714093 
## converged
## initial  value 10.725882 
## iter   2 value 10.724655
## iter   3 value 10.724042
## iter   4 value 10.723563
## iter   5 value 10.723465
## iter   6 value 10.723337
## iter   7 value 10.722543
## iter   8 value 10.721962
## iter   9 value 10.720987
## iter  10 value 10.719763
## iter  11 value 10.718355
## iter  12 value 10.717664
## iter  13 value 10.717299
## iter  14 value 10.717109
## iter  15 value 10.716965
## iter  16 value 10.716894
## iter  17 value 10.716859
## iter  18 value 10.716723
## iter  19 value 10.716685
## iter  20 value 10.716650
## iter  21 value 10.716639
## iter  22 value 10.716518
## iter  23 value 10.716410
## iter  24 value 10.716162
## iter  25 value 10.715816
## iter  26 value 10.715631
## iter  27 value 10.715531
## iter  28 value 10.715509
## iter  29 value 10.715494
## iter  30 value 10.715476
## iter  31 value 10.715447
## iter  32 value 10.715404
## iter  33 value 10.715371
## iter  34 value 10.715368
## iter  35 value 10.715367
## iter  35 value 10.715367
## final  value 10.715367 
## converged

## $fit
## 
## Call:
## stats::arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, 
##     Q), period = S), include.mean = !no.constant, transform.pars = trans, fixed = fixed, 
##     optim.control = list(trace = trc, REPORT = 1, reltol = tol))
## 
## Coefficients:
##          ar1     ar2     ma1      ma2      ma3     sar1     sar2     sma1
##       0.0434  0.8824  0.2511  -0.7471  -0.2454  -0.1288  -0.0045  -0.8871
## s.e.  0.0791  0.0754  0.0932   0.0862   0.0553   0.0721   0.0681   0.0483
## 
## sigma^2 estimated as 1.903e+09:  log likelihood = -3798.04,  aic = 7614.08
## 
## $degrees_of_freedom
## [1] 305
## 
## $ttable
##      Estimate     SE  t.value p.value
## ar1    0.0434 0.0791   0.5489  0.5835
## ar2    0.8824 0.0754  11.7013  0.0000
## ma1    0.2511 0.0932   2.6946  0.0074
## ma2   -0.7471 0.0862  -8.6637  0.0000
## ma3   -0.2454 0.0553  -4.4386  0.0000
## sar1  -0.1288 0.0721  -1.7859  0.0751
## sar2  -0.0045 0.0681  -0.0657  0.9476
## sma1  -0.8871 0.0483 -18.3607  0.0000
## 
## $AIC
## [1] 23.50023
## 
## $AICc
## [1] 23.50164
## 
## $BIC
## [1] 23.60429

Prediction

pred22 = forecast(fit22,h=48)
pred22

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Jun 2019       32864165 32807113 32921218 32776911 32951420
## Jul 2019       32983157 32889832 33076482 32840429 33125885
## Aug 2019       33021524 32897095 33145954 32831225 33211823
## Sep 2019       33024187 32874342 33174031 32795020 33253354
## Oct 2019       33051291 32875983 33226599 32783181 33319402
## Nov 2019       33087272 32889119 33285425 32784223 33390321
## Dec 2019       33061521 32839992 33283050 32722721 33400321
## Jan 2020       33025561 32782279 33268842 32653494 33397627
## Feb 2020       33032692 32767117 33298267 32626530 33438854
## Mar 2020       33070450 32783719 33357182 32631932 33508968
## Apr 2020       33099202 32790852 33407551 32627621 33570782
## May 2020       33135923 32806808 33465037 32632585 33639260
## Jun 2020       33232482 32882538 33582427 32697289 33767676
## Jul 2020       33347528 32977479 33717576 32781587 33913469
## Aug 2020       33379790 32989372 33770209 32782696 33976885
## Sep 2020       33384722 32974450 33794994 32757265 34012179
## Oct 2020       33414795 32984476 33845114 32756679 34072911
## Nov 2020       33454453 33004519 33904388 32766338 34142569
## Dec 2020       33438402 32968719 33908085 32720084 34156720
## Jan 2021       33396378 32907314 33885441 32648420 34144336
## Feb 2021       33398803 32890276 33907330 32621077 34176528
## Mar 2021       33437801 32910129 33965472 32630797 34244804
## Apr 2021       33461260 32914402 34008117 32624913 34297606
## May 2021       33499523 32933763 34065283 32634267 34364779
## Jun 2021       33596477 33010179 34182775 32699812 34493142
## Jul 2021       33711166 33104196 34318136 32782885 34639447
## Aug 2021       33742789 33115022 34370555 32782703 34702875
## Sep 2021       33746812 33098558 34395067 32755392 34738233
## Oct 2021       33775487 33106671 34444303 32752621 34798352
## Nov 2021       33814150 33125051 34503250 32760263 34868037
## Dec 2021       33796183 33086768 34505598 32711226 34881140
## Jan 2022       33754096 33024619 34483574 32638457 34869736
## Feb 2022       33756032 33006493 34505571 32609711 34902353
## Mar 2022       33794318 33024951 34563685 32617672 34970964
## Apr 2022       33817439 33028271 34606607 32610511 35024368
## May 2022       33855005 33046253 34663756 32618126 35091884
## Jun 2022       33951166 33121642 34780691 32682518 35219814
## Jul 2022       34065363 33214953 34915772 32764774 35365951
## Aug 2022       34096355 33225001 34967709 32763734 35428976
## Sep 2022       34099963 33207893 34992032 32735660 35464265
## Oct 2022       34128128 33215321 35040935 32732110 35524146
## Nov 2022       34166413 33233082 35099743 32739007 35593818
## Dec 2022       34148031 33194185 35101877 32689250 35606812
## Jan 2023       34105520 33131361 35079679 32615672 35595368
## Feb 2023       34106975 33112534 35101415 32586109 35627840
## Mar 2023       34144916 33130386 35159447 32593325 35696507
## Apr 2023       34167588 33133018 35202157 32585350 35749825
## May 2023       34204834 33150409 35259259 32592230 35817438

autoplot(pred22)

4. ARCH/GARCH model

# fit ARIMA model without seasonal term 
fit3=auto.arima(df22,  seasonal = FALSE)
fit3

## Series: df22 
## ARIMA(5,1,1) with drift 
## 
## Coefficients:
##          ar1      ar2      ar3     ar4      ar5      ma1      drift
##       0.7583  -0.2536  -0.0473  0.2995  -0.4508  -0.2670  22452.337
## s.e.  0.0835   0.0718   0.0652  0.0631   0.0500   0.0853   2975.167
## 
## sigma^2 estimated as 2.608e+09:  log likelihood=-3981.5
## AIC=7979   AICc=7979.46   BIC=8009.27

## ARIMA(5,1,1)

df22.diff=diff(df22)
plot(df22.diff, main=" diff data")

df22.log=log(df22)
plot(df22.log, main=" log data")

df22.difflog=diff(log(df22))
plot(df22.difflog, main=" difflog data")

acf2(df22.difflog)

##         ACF  PACF
##  [1,]  0.47  0.47
##  [2,]  0.11 -0.14
##  [3,]  0.00  0.02
##  [4,]  0.04  0.06
##  [5,] -0.29 -0.44
##  [6,] -0.47 -0.21
##  [7,] -0.27  0.09
##  [8,]  0.02  0.11
##  [9,] -0.01 -0.07
## [10,]  0.09  0.20
## [11,]  0.32  0.15
## [12,]  0.50  0.17
## [13,]  0.29  0.04
## [14,]  0.01 -0.15
## [15,] -0.03  0.02
## [16,]  0.00  0.10
## [17,] -0.25 -0.15
## [18,] -0.44 -0.11
## [19,] -0.30 -0.02
## [20,]  0.00  0.04
## [21,] -0.05 -0.15
## [22,]  0.01  0.11
## [23,]  0.31  0.18
## [24,]  0.51  0.07
## [25,]  0.30  0.03
## [26,]  0.05  0.00
## [27,]  0.08  0.10
## [28,]  0.02 -0.07
## [29,] -0.27 -0.05
## [30,] -0.46 -0.09
## [31,] -0.28  0.02
## [32,] -0.06 -0.01
## [33,] -0.06 -0.06
## [34,]  0.00  0.03
## [35,]  0.29  0.06
## [36,]  0.45  0.01
## [37,]  0.28  0.02
## [38,]  0.05  0.01
## [39,]  0.06  0.02
## [40,]  0.01 -0.03
## [41,] -0.29 -0.08
## [42,] -0.47 -0.04
## [43,] -0.29 -0.02
## [44,] -0.06  0.03
## [45,] -0.05  0.00
## [46,] -0.02 -0.01
## [47,]  0.27  0.01
## [48,]  0.42  0.01

## From the acf and pacf we found that there is some patterns. Therefore, we should fit a garch model.  AR(1)
acf2(df22^2)## ARCH(1)

##        ACF  PACF
##  [1,] 0.99  0.99
##  [2,] 0.98 -0.02
##  [3,] 0.97 -0.01
##  [4,] 0.96  0.01
##  [5,] 0.95 -0.01
##  [6,] 0.94 -0.02
##  [7,] 0.92  0.01
##  [8,] 0.91  0.00
##  [9,] 0.90 -0.01
## [10,] 0.89 -0.02
## [11,] 0.88 -0.02
## [12,] 0.87 -0.01
## [13,] 0.86 -0.01
## [14,] 0.85 -0.02
## [15,] 0.84  0.00
## [16,] 0.82  0.00
## [17,] 0.81  0.01
## [18,] 0.80 -0.02
## [19,] 0.79  0.01
## [20,] 0.78 -0.01
## [21,] 0.77  0.00
## [22,] 0.76 -0.02
## [23,] 0.74 -0.02
## [24,] 0.73 -0.01
## [25,] 0.72 -0.01
## [26,] 0.71 -0.02
## [27,] 0.70  0.01
## [28,] 0.69  0.00
## [29,] 0.68  0.00
## [30,] 0.66 -0.01
## [31,] 0.65  0.00
## [32,] 0.64  0.01
## [33,] 0.63 -0.02
## [34,] 0.62 -0.02
## [35,] 0.61 -0.01
## [36,] 0.60  0.00
## [37,] 0.59 -0.01
## [38,] 0.57 -0.01
## [39,] 0.56  0.01
## [40,] 0.55  0.00
## [41,] 0.54 -0.01
## [42,] 0.53 -0.01
## [43,] 0.52  0.00
## [44,] 0.51  0.01
## [45,] 0.50 -0.01
## [46,] 0.49 -0.01
## [47,] 0.48  0.00
## [48,] 0.47  0.00

# fit arima(5,1,1)
arima511 = arima(df22,order=c(5,1,1))
residual_arima511 = arima511$residuals
acf2(residual_arima511^2)

##         ACF  PACF
##  [1,] -0.01 -0.01
##  [2,]  0.20  0.20
##  [3,]  0.07  0.07
##  [4,]  0.07  0.03
##  [5,]  0.03  0.01
##  [6,]  0.02  0.00
##  [7,] -0.02 -0.03
##  [8,]  0.02  0.01
##  [9,] -0.07 -0.07
## [10,]  0.13  0.12
## [11,]  0.04  0.07
## [12,]  0.09  0.06
## [13,]  0.07  0.04
## [14,]  0.06  0.02
## [15,]  0.06  0.02
## [16,] -0.07 -0.11
## [17,]  0.15  0.14
## [18,] -0.02  0.00
## [19,] -0.02 -0.05
## [20,]  0.01  0.00
## [21,]  0.03  0.05
## [22,]  0.04  0.04
## [23,]  0.01 -0.02
## [24,]  0.00 -0.03
## [25,]  0.09  0.06
## [26,] -0.04 -0.01
## [27,]  0.01 -0.06
## [28,] -0.08 -0.10
## [29,] -0.02 -0.02
## [30,]  0.02  0.07
## [31,]  0.03  0.06
## [32,] -0.05 -0.08
## [33,]  0.03  0.03
## [34,] -0.07 -0.08
## [35,]  0.14  0.11
## [36,] -0.04 -0.01
## [37,]  0.06  0.02
## [38,]  0.01  0.02
## [39,]  0.03  0.01
## [40,]  0.02  0.03
## [41,] -0.01 -0.01
## [42,]  0.07  0.06
## [43,]  0.17  0.19
## [44,]  0.00  0.02
## [45,]  0.23  0.19
## [46,]  0.05  0.05
## [47,]  0.15  0.05
## [48,]  0.12  0.07

summary(garchFit(~arma(5,1)+garch(1,1), df22.difflog))

## 
## Series Initialization:
##  ARMA Model:                arma
##  Formula Mean:              ~ arma(5, 1)
##  GARCH Model:               garch
##  Formula Variance:          ~ garch(1, 1)
##  ARMA Order:                5 1
##  Max ARMA Order:            5
##  GARCH Order:               1 1
##  Max GARCH Order:           1
##  Maximum Order:             5
##  Conditional Dist:          norm
##  h.start:                   6
##  llh.start:                 1
##  Length of Series:          325
##  Recursion Init:            mci
##  Series Scale:              0.002262591
## 
## Parameter Initialization:
##  Initial Parameters:          $params
##  Limits of Transformations:   $U, $V
##  Which Parameters are Fixed?  $includes
##  Parameter Matrix:
##                      U          V      params includes
##     mu     -3.35797923   3.357979  0.34224892     TRUE
##     ar1    -0.99999999   1.000000  0.76916698     TRUE
##     ar2    -0.99999999   1.000000 -0.26344089     TRUE
##     ar3    -0.99999999   1.000000 -0.04869207     TRUE
##     ar4    -0.99999999   1.000000  0.31133629     TRUE
##     ar5    -0.99999999   1.000000 -0.47352930     TRUE
##     ma1    -0.99999999   1.000000 -0.26786887     TRUE
##     omega   0.00000100 100.000000  0.10000000     TRUE
##     alpha1  0.00000001   1.000000  0.10000000     TRUE
##     gamma1 -0.99999999   1.000000  0.10000000    FALSE
##     beta1   0.00000001   1.000000  0.80000000     TRUE
##     delta   0.00000000   2.000000  2.00000000    FALSE
##     skew    0.10000000  10.000000  1.00000000    FALSE
##     shape   1.00000000  10.000000  4.00000000    FALSE
##  Index List of Parameters to be Optimized:
##     mu    ar1    ar2    ar3    ar4    ar5    ma1  omega alpha1  beta1 
##      1      2      3      4      5      6      7      8      9     11 
##  Persistence:                  0.9 
## 
## 
## --- START OF TRACE ---
## Selected Algorithm: nlminb 
## 
## R coded nlminb Solver: 
## 
##   0:     377.48613: 0.342249 0.769167 -0.263441 -0.0486921 0.311336 -0.473529 -0.267869 0.100000 0.100000 0.800000
##   1:     373.43253: 0.341966 0.767989 -0.264698 -0.0500675 0.309883 -0.475060 -0.268039 0.0914690 0.0955537 0.793343
##   2:     369.77578: 0.341175 0.764696 -0.268163 -0.0538182 0.305889 -0.479355 -0.268624 0.0744947 0.0866795 0.781362
##   3:     368.73389: 0.339117 0.756345 -0.276560 -0.0626328 0.296259 -0.490454 -0.270910 0.0834987 0.0900083 0.785699
##   4:     368.03044: 0.338569 0.754972 -0.277630 -0.0636957 0.294948 -0.492639 -0.271650 0.0745500 0.0858335 0.780094
##   5:     367.70195: 0.336522 0.751671 -0.278907 -0.0644204 0.293149 -0.499261 -0.275301 0.0785891 0.0894200 0.781818
##   6:     367.28922: 0.331155 0.748399 -0.277145 -0.0609421 0.295413 -0.510728 -0.284619 0.0715862 0.0915855 0.778275
##   7:     366.94931: 0.324740 0.751911 -0.275357 -0.0592148 0.302971 -0.513841 -0.293392 0.0770102 0.0950558 0.776720
##   8:     366.50452: 0.312026 0.771518 -0.280474 -0.0740057 0.309454 -0.507479 -0.304730 0.0780596 0.0932281 0.770608
##   9:     366.20279: 0.299707 0.780180 -0.274370 -0.0580949 0.310622 -0.514114 -0.328066 0.0815709 0.0917378 0.770731
##  10:     366.14197: 0.299443 0.779615 -0.275475 -0.0591049 0.310093 -0.514353 -0.328252 0.0793892 0.0910641 0.769552
##  11:     366.10758: 0.299022 0.778908 -0.277049 -0.0604938 0.309471 -0.514591 -0.328577 0.0805255 0.0922177 0.770262
##  12:     366.09882: 0.298644 0.778663 -0.278069 -0.0613373 0.309291 -0.514536 -0.328812 0.0782208 0.0917513 0.769054
##  13:     366.06346: 0.298101 0.778455 -0.279054 -0.0609810 0.309583 -0.513873 -0.329487 0.0791906 0.0936049 0.770073
##  14:     366.03901: 0.296698 0.778852 -0.281361 -0.0598142 0.309983 -0.512431 -0.330812 0.0764715 0.0958237 0.770026
##  15:     365.99987: 0.294628 0.786842 -0.285624 -0.0568315 0.306066 -0.512029 -0.330173 0.0756710 0.0976599 0.773711
##  16:     365.94812: 0.292484 0.789228 -0.291192 -0.0537997 0.306113 -0.508544 -0.335023 0.0764872 0.0915949 0.775345
##  17:     365.94001: 0.291949 0.789797 -0.291251 -0.0545509 0.305967 -0.508933 -0.335522 0.0770443 0.0925668 0.775575
##  18:     365.92707: 0.291604 0.790507 -0.290584 -0.0553598 0.305807 -0.508879 -0.336400 0.0759213 0.0927467 0.774935
##  19:     365.91671: 0.291154 0.791021 -0.289983 -0.0559237 0.306175 -0.508565 -0.337516 0.0763639 0.0936045 0.775066
##  20:     365.89750: 0.290100 0.792285 -0.289076 -0.0567057 0.307640 -0.508215 -0.339503 0.0754878 0.0941219 0.774363
##  21:     365.88368: 0.288182 0.798276 -0.289805 -0.0579616 0.306424 -0.511027 -0.339172 0.0765511 0.0930184 0.774667
##  22:     365.84244: 0.286201 0.796859 -0.292015 -0.0559792 0.309535 -0.507427 -0.342878 0.0758738 0.0927686 0.774425
##  23:     365.83730: 0.286107 0.796930 -0.292060 -0.0561294 0.309444 -0.507540 -0.342985 0.0765135 0.0931136 0.774745
##  24:     365.83418: 0.285903 0.797120 -0.292115 -0.0564029 0.309308 -0.507728 -0.343204 0.0761681 0.0930633 0.774487
##  25:     365.82944: 0.285523 0.797509 -0.292042 -0.0562667 0.308742 -0.507346 -0.344080 0.0766514 0.0932967 0.774801
##  26:     365.81938: 0.284696 0.798532 -0.292251 -0.0552691 0.307994 -0.506398 -0.345578 0.0763433 0.0927186 0.774557
##  27:     365.80708: 0.283245 0.800972 -0.294728 -0.0540023 0.311513 -0.508379 -0.345610 0.0768962 0.0932732 0.774440
##  28:     365.78108: 0.281193 0.803328 -0.292120 -0.0569490 0.307733 -0.503450 -0.355235 0.0752569 0.0945944 0.776247
##  29:     365.77861: 0.281116 0.803507 -0.292154 -0.0570100 0.307721 -0.503613 -0.355185 0.0748725 0.0943438 0.775964
##  30:     365.77668: 0.280996 0.803777 -0.292210 -0.0570921 0.307728 -0.503846 -0.355123 0.0752300 0.0943957 0.776051
##  31:     365.77410: 0.280676 0.804006 -0.292074 -0.0571378 0.307569 -0.503683 -0.354749 0.0750979 0.0938830 0.775627
##  32:     365.77009: 0.280057 0.804055 -0.291686 -0.0570893 0.307082 -0.502760 -0.353959 0.0760862 0.0934396 0.775341
##  33:     365.75550: 0.279191 0.809881 -0.298032 -0.0564349 0.310382 -0.505257 -0.361202 0.0760220 0.0957415 0.773421
##  34:     365.74772: 0.275147 0.811142 -0.297880 -0.0549356 0.310227 -0.502986 -0.359403 0.0752504 0.0947386 0.772444
##  35:     365.72846: 0.272515 0.814620 -0.296982 -0.0565074 0.312663 -0.502523 -0.365053 0.0768724 0.0897596 0.776635
##  36:     365.72107: 0.272146 0.816441 -0.296757 -0.0574921 0.310950 -0.503976 -0.363591 0.0757229 0.0920860 0.776352
##  37:     365.72020: 0.272131 0.816451 -0.296744 -0.0573734 0.311130 -0.503803 -0.363590 0.0759592 0.0922544 0.776461
##  38:     365.71972: 0.272105 0.816452 -0.296752 -0.0572597 0.311316 -0.503620 -0.363586 0.0757171 0.0922106 0.776296
##  39:     365.71897: 0.272030 0.816441 -0.296892 -0.0569999 0.311178 -0.503415 -0.363547 0.0761254 0.0925374 0.775821
##  40:     365.71819: 0.271866 0.816420 -0.297197 -0.0564421 0.310773 -0.503046 -0.363452 0.0764201 0.0929133 0.774551
##  41:     365.71715: 0.271403 0.817112 -0.298075 -0.0559646 0.309982 -0.502116 -0.363317 0.0775007 0.0937605 0.772452
##  42:     365.71601: 0.270668 0.817371 -0.297798 -0.0563674 0.310964 -0.502108 -0.363515 0.0765175 0.0932707 0.774421
##  43:     365.71601: 0.270282 0.818869 -0.298435 -0.0550688 0.310548 -0.501631 -0.365253 0.0770952 0.0927087 0.773217
##  44:     365.71580: 0.269914 0.818127 -0.298041 -0.0547517 0.310487 -0.502029 -0.365321 0.0777691 0.0929483 0.772890
##  45:     365.71502: 0.270024 0.818317 -0.297846 -0.0559215 0.311166 -0.501960 -0.365159 0.0779606 0.0929667 0.772045
##  46:     365.71495: 0.269910 0.818774 -0.298692 -0.0553834 0.310578 -0.501664 -0.365278 0.0773870 0.0928270 0.772861
##  47:     365.71483: 0.269926 0.819242 -0.298979 -0.0555634 0.310613 -0.501483 -0.365396 0.0774700 0.0929646 0.773050
##  48:     365.71468: 0.269850 0.819032 -0.298701 -0.0556750 0.310811 -0.501648 -0.365339 0.0775362 0.0931294 0.772651
##  49:     365.71466: 0.269456 0.818731 -0.298424 -0.0553742 0.310926 -0.501804 -0.365383 0.0777759 0.0930693 0.772449
##  50:     365.71450: 0.269284 0.819054 -0.298573 -0.0553978 0.310997 -0.501701 -0.365556 0.0777113 0.0930174 0.772387
##  51:     365.71444: 0.269199 0.819276 -0.298538 -0.0556780 0.311212 -0.501719 -0.365640 0.0778228 0.0932844 0.772048
##  52:     365.71441: 0.269084 0.819555 -0.298811 -0.0554462 0.311086 -0.501582 -0.365770 0.0777411 0.0931977 0.772282
##  53:     365.71439: 0.269041 0.819575 -0.298898 -0.0555197 0.311127 -0.501534 -0.365705 0.0777617 0.0931477 0.772234
##  54:     365.71439: 0.269054 0.819686 -0.298983 -0.0554234 0.311061 -0.501499 -0.365730 0.0777118 0.0932094 0.772243
##  55:     365.71438: 0.269030 0.819808 -0.299099 -0.0553812 0.311026 -0.501451 -0.365790 0.0777342 0.0932086 0.772236
##  56:     365.71438: 0.269022 0.819824 -0.299133 -0.0553754 0.311030 -0.501456 -0.365809 0.0777028 0.0931964 0.772300
##  57:     365.71438: 0.269020 0.819832 -0.299130 -0.0553763 0.311028 -0.501452 -0.365821 0.0777140 0.0931971 0.772279
##  58:     365.71438: 0.269022 0.819824 -0.299126 -0.0553776 0.311030 -0.501453 -0.365810 0.0777117 0.0931978 0.772283
## 
## Final Estimate of the Negative LLH:
##  LLH:  -1613.94    norm LLH:  -4.96597 
##            mu           ar1           ar2           ar3           ar4 
##  6.086864e-04  8.198240e-01 -2.991259e-01 -5.537759e-02  3.110303e-01 
##           ar5           ma1         omega        alpha1         beta1 
## -5.014525e-01 -3.658103e-01  3.978309e-07  9.319782e-02  7.722835e-01 
## 
## R-optimhess Difference Approximated Hessian Matrix:
##                   mu           ar1           ar2           ar3
## mu     -2.925216e+08 -2.847106e+05 -2.784705e+05 -266841.99714
## ar1    -2.847106e+05 -1.100574e+03 -8.070901e+02    -434.27976
## ar2    -2.784705e+05 -8.070901e+02 -1.073956e+03    -778.51528
## ar3    -2.668420e+05 -4.342798e+02 -7.785153e+02   -1097.67814
## ar4    -2.703049e+05 -2.169215e+02 -4.214022e+02    -822.30807
## ar5    -2.634540e+05 -7.711196e+01 -2.095625e+02    -444.56051
## ma1    -3.315680e+04 -3.778250e+02 -1.345091e+02     -31.59150
## omega  -1.369639e+10  1.673603e+07  1.542630e+07 9350385.84576
## alpha1  2.657556e+04 -2.208810e+01 -4.232767e+01     -34.97202
## beta1  -3.069864e+04  3.809996e+01  3.061092e+01      17.60326
##                  ar4           ar5           ma1         omega
## mu     -2.703049e+05 -2.634540e+05 -3.315680e+04 -1.369639e+10
## ar1    -2.169215e+02 -7.711196e+01 -3.778250e+02  1.673603e+07
## ar2    -4.214022e+02 -2.095625e+02 -1.345091e+02  1.542630e+07
## ar3    -8.223081e+02 -4.445605e+02 -3.159150e+01  9.350386e+06
## ar4    -1.152946e+03 -8.433250e+02 -7.655507e+00  1.193937e+07
## ar5    -8.433250e+02 -1.141962e+03 -1.076582e+01  2.370733e+07
## ma1    -7.655507e+00 -1.076582e+01 -3.519323e+02  2.753895e+07
## omega   1.193937e+07  2.370733e+07  2.753895e+07 -4.194480e+14
## alpha1 -3.997496e+01 -7.811831e+01 -5.989128e+01 -9.153461e+08
## beta1   2.849936e+01  2.834734e+01  4.388268e+01 -1.159172e+09
##               alpha1         beta1
## mu      2.657556e+04 -3.069864e+04
## ar1    -2.208810e+01  3.809996e+01
## ar2    -4.232767e+01  3.061092e+01
## ar3    -3.497202e+01  1.760326e+01
## ar4    -3.997496e+01  2.849936e+01
## ar5    -7.811831e+01  2.834734e+01
## ma1    -5.989128e+01  4.388268e+01
## omega  -9.153461e+08 -1.159172e+09
## alpha1 -2.933753e+03 -2.767539e+03
## beta1  -2.767539e+03 -3.334359e+03
## attr(,"time")
## Time difference of 0.02720594 secs
## 
## --- END OF TRACE ---
## 
## 
## Time to Estimate Parameters:
##  Time difference of 0.121706 secs
## 
## Title:
##  GARCH Modelling 
## 
## Call:
##  garchFit(formula = ~arma(5, 1) + garch(1, 1), data = df22.difflog) 
## 
## Mean and Variance Equation:
##  data ~ arma(5, 1) + garch(1, 1)
## <environment: 0x7fafd724b790>
##  [data = df22.difflog]
## 
## Conditional Distribution:
##  norm 
## 
## Coefficient(s):
##          mu          ar1          ar2          ar3          ar4  
##  6.0869e-04   8.1982e-01  -2.9913e-01  -5.5378e-02   3.1103e-01  
##         ar5          ma1        omega       alpha1        beta1  
## -5.0145e-01  -3.6581e-01   3.9783e-07   9.3198e-02   7.7228e-01  
## 
## Std. Errors:
##  based on Hessian 
## 
## Error Analysis:
##          Estimate  Std. Error  t value Pr(>|t|)    
## mu      6.087e-04   8.471e-05    7.186 6.69e-13 ***
## ar1     8.198e-01   7.165e-02   11.442  < 2e-16 ***
## ar2    -2.991e-01   7.201e-02   -4.154 3.27e-05 ***
## ar3    -5.538e-02   6.812e-02   -0.813   0.4163    
## ar4     3.110e-01   6.510e-02    4.777 1.78e-06 ***
## ar5    -5.015e-01   5.000e-02  -10.029  < 2e-16 ***
## ma1    -3.658e-01   8.055e-02   -4.541 5.59e-06 ***
## omega   3.978e-07   2.806e-07    1.418   0.1563    
## alpha1  9.320e-02   4.655e-02    2.002   0.0453 *  
## beta1   7.723e-01   1.202e-01    6.424 1.33e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log Likelihood:
##  1613.94    normalized:  4.96597 
## 
## Description:
##  Thu Apr 30 16:55:59 2020 by user:  
## 
## 
## Standardised Residuals Tests:
##                                 Statistic p-Value     
##  Jarque-Bera Test   R    Chi^2  0.4475601 0.799491    
##  Shapiro-Wilk Test  R    W      0.9965797 0.7184867   
##  Ljung-Box Test     R    Q(10)  35.19155  0.0001158062
##  Ljung-Box Test     R    Q(15)  49.83029  1.28346e-05 
##  Ljung-Box Test     R    Q(20)  61.18389  4.663229e-06
##  Ljung-Box Test     R^2  Q(10)  3.870005  0.9530208   
##  Ljung-Box Test     R^2  Q(15)  4.625821  0.9948484   
##  Ljung-Box Test     R^2  Q(20)  16.90143  0.6593643   
##  LM Arch Test       R    TR^2   4.752299  0.9657447   
## 
## Information Criterion Statistics:
##       AIC       BIC       SIC      HQIC 
## -9.870401 -9.753975 -9.872220 -9.823935

summary(garchFit(~arma(5,1)+garch(1,0), df22.difflog))

## 
## Series Initialization:
##  ARMA Model:                arma
##  Formula Mean:              ~ arma(5, 1)
##  GARCH Model:               garch
##  Formula Variance:          ~ garch(1, 0)
##  ARMA Order:                5 1
##  Max ARMA Order:            5
##  GARCH Order:               1 0
##  Max GARCH Order:           1
##  Maximum Order:             5
##  Conditional Dist:          norm
##  h.start:                   6
##  llh.start:                 1
##  Length of Series:          325
##  Recursion Init:            mci
##  Series Scale:              0.002262591
## 
## Parameter Initialization:
##  Initial Parameters:          $params
##  Limits of Transformations:   $U, $V
##  Which Parameters are Fixed?  $includes
##  Parameter Matrix:
##                      U          V      params includes
##     mu     -3.35797923   3.357979  0.34224892     TRUE
##     ar1    -0.99999999   1.000000  0.76916698     TRUE
##     ar2    -0.99999999   1.000000 -0.26344089     TRUE
##     ar3    -0.99999999   1.000000 -0.04869207     TRUE
##     ar4    -0.99999999   1.000000  0.31133629     TRUE
##     ar5    -0.99999999   1.000000 -0.47352930     TRUE
##     ma1    -0.99999999   1.000000 -0.26786887     TRUE
##     omega   0.00000100 100.000000  0.10000000     TRUE
##     alpha1  0.00000001   1.000000  0.10000000     TRUE
##     gamma1 -0.99999999   1.000000  0.10000000    FALSE
##     delta   0.00000000   2.000000  2.00000000    FALSE
##     skew    0.10000000  10.000000  1.00000000    FALSE
##     shape   1.00000000  10.000000  4.00000000    FALSE
##  Index List of Parameters to be Optimized:
##     mu    ar1    ar2    ar3    ar4    ar5    ma1  omega alpha1 
##      1      2      3      4      5      6      7      8      9 
##  Persistence:                  0.1 
## 
## 
## --- START OF TRACE ---
## Selected Algorithm: nlminb 
## 
## R coded nlminb Solver: 
## 
##   0:     656.92593: 0.342249 0.769167 -0.263441 -0.0486921 0.311336 -0.473529 -0.267869 0.100000 0.100000
##   1:     411.39194: 0.334326 0.719643 -0.309220 -0.0938436 0.278791 -0.505450 -0.280305  1.06289 0.351924
##   2:     407.29115: 0.334049 0.745665 -0.273313 -0.0606229 0.311782 -0.482879 -0.268738  1.04190 0.332778
##   3:     387.88711: 0.320198 0.724967 -0.296579 -0.101467 0.276034 -0.519209 -0.266348 0.780927 0.211536
##   4:     381.50468: 0.321064 0.745784 -0.265518 -0.0681479 0.311563 -0.490873 -0.258885 0.741282 0.192545
##   5:     372.94902: 0.312822 0.731338 -0.276035 -0.0833404 0.304037 -0.496966 -0.265312 0.598536 0.124656
##   6:     371.73803: 0.311627 0.746773 -0.255515 -0.0654637 0.321182 -0.484252 -0.261429 0.572152 0.114446
##   7:     371.00952: 0.302306 0.744145 -0.260390 -0.0829683 0.302904 -0.499720 -0.262409 0.548801 0.108482
##   8:     370.38398: 0.292004 0.740892 -0.256099 -0.0746217 0.324622 -0.477351 -0.271440 0.535843 0.109528
##   9:     370.13521: 0.281298 0.764679 -0.244808 -0.0797040 0.313179 -0.491924 -0.273997 0.522562 0.110399
##  10:     369.92133: 0.279562 0.752659 -0.255774 -0.0809292 0.321060 -0.481661 -0.281635 0.518408 0.106459
##  11:     369.85168: 0.274403 0.768501 -0.252858 -0.0771533 0.322529 -0.482120 -0.287386 0.516276 0.105139
##  12:     369.80973: 0.268216 0.772768 -0.260184 -0.0817655 0.317428 -0.485790 -0.296173 0.515065 0.104321
##  13:     369.70064: 0.262850 0.780875 -0.263244 -0.0733996 0.319989 -0.477153 -0.303659 0.517708 0.101692
##  14:     369.69037: 0.257572 0.793623 -0.269156 -0.0760200 0.318429 -0.475426 -0.309900 0.520281 0.0946845
##  15:     369.67884: 0.256094 0.791242 -0.268669 -0.0703742 0.320819 -0.475390 -0.315068 0.520360 0.0942031
##  16:     369.67677: 0.255885 0.790859 -0.269799 -0.0721397 0.318831 -0.476551 -0.314756 0.520239 0.0944700
##  17:     369.67263: 0.255676 0.790991 -0.268856 -0.0712070 0.318719 -0.474875 -0.313416 0.519460 0.0962086
##  18:     369.67245: 0.255341 0.791038 -0.268985 -0.0710281 0.318179 -0.474949 -0.313678 0.519150 0.0966565
##  19:     369.67235: 0.255150 0.791586 -0.269267 -0.0707610 0.318561 -0.474723 -0.314371 0.518934 0.0971124
##  20:     369.67226: 0.255131 0.791600 -0.269452 -0.0710565 0.318386 -0.474717 -0.314288 0.518879 0.0970360
##  21:     369.67220: 0.255090 0.791755 -0.269297 -0.0709599 0.318432 -0.474568 -0.314210 0.518943 0.0967813
##  22:     369.67217: 0.255011 0.791762 -0.269364 -0.0709838 0.318336 -0.474655 -0.314217 0.519050 0.0964945
##  23:     369.67216: 0.254962 0.791725 -0.269409 -0.0709866 0.318389 -0.474358 -0.314086 0.519167 0.0963370
##  24:     369.67212: 0.254893 0.791945 -0.269350 -0.0710019 0.318322 -0.474511 -0.314235 0.519208 0.0961793
##  25:     369.67210: 0.254838 0.792110 -0.269609 -0.0709493 0.318464 -0.474463 -0.314386 0.519171 0.0962208
##  26:     369.67210: 0.254808 0.792182 -0.269564 -0.0710164 0.318300 -0.474453 -0.314457 0.519162 0.0962504
##  27:     369.67209: 0.254771 0.792225 -0.269469 -0.0709343 0.318308 -0.474341 -0.314527 0.519196 0.0961856
##  28:     369.67208: 0.254717 0.792171 -0.269519 -0.0708951 0.318278 -0.474402 -0.314411 0.519225 0.0961578
##  29:     369.67208: 0.254705 0.792197 -0.269556 -0.0709168 0.318312 -0.474341 -0.314438 0.519203 0.0961521
##  30:     369.67208: 0.254692 0.792244 -0.269568 -0.0709421 0.318270 -0.474364 -0.314494 0.519202 0.0961522
##  31:     369.67208: 0.254678 0.792307 -0.269581 -0.0708962 0.318305 -0.474364 -0.314509 0.519205 0.0961360
##  32:     369.67208: 0.254665 0.792314 -0.269601 -0.0708950 0.318271 -0.474332 -0.314584 0.519213 0.0961249
##  33:     369.67207: 0.254645 0.792378 -0.269619 -0.0708968 0.318296 -0.474329 -0.314568 0.519209 0.0961594
##  34:     369.67207: 0.254638 0.792362 -0.269651 -0.0709136 0.318288 -0.474346 -0.314584 0.519204 0.0961615
##  35:     369.67207: 0.254630 0.792384 -0.269646 -0.0708853 0.318291 -0.474333 -0.314606 0.519208 0.0961474
##  36:     369.67207: 0.254618 0.792393 -0.269669 -0.0708953 0.318292 -0.474324 -0.314627 0.519200 0.0961587
##  37:     369.67207: 0.254596 0.792456 -0.269703 -0.0708638 0.318297 -0.474347 -0.314639 0.519199 0.0961524
##  38:     369.67207: 0.254595 0.792449 -0.269699 -0.0708570 0.318313 -0.474315 -0.314657 0.519200 0.0961550
##  39:     369.67207: 0.254586 0.792449 -0.269689 -0.0708720 0.318296 -0.474320 -0.314669 0.519206 0.0961436
##  40:     369.67207: 0.254588 0.792468 -0.269698 -0.0708561 0.318279 -0.474306 -0.314683 0.519197 0.0961587
##  41:     369.67207: 0.254584 0.792469 -0.269688 -0.0708824 0.318298 -0.474316 -0.314695 0.519193 0.0961718
##  42:     369.67207: 0.254579 0.792475 -0.269711 -0.0708638 0.318288 -0.474304 -0.314707 0.519198 0.0961588
##  43:     369.67207: 0.254575 0.792482 -0.269701 -0.0708615 0.318290 -0.474303 -0.314706 0.519201 0.0961557
##  44:     369.67207: 0.254577 0.792485 -0.269709 -0.0708620 0.318279 -0.474308 -0.314703 0.519197 0.0961655
##  45:     369.67207: 0.254575 0.792487 -0.269709 -0.0708613 0.318284 -0.474305 -0.314707 0.519197 0.0961645
##  46:     369.67207: 0.254573 0.792490 -0.269710 -0.0708615 0.318287 -0.474303 -0.314711 0.519196 0.0961625
##  47:     369.67207: 0.254568 0.792492 -0.269710 -0.0708611 0.318286 -0.474303 -0.314719 0.519201 0.0961555
##  48:     369.67207: 0.254568 0.792504 -0.269717 -0.0708591 0.318288 -0.474304 -0.314725 0.519198 0.0961646
##  49:     369.67207: 0.254567 0.792503 -0.269718 -0.0708602 0.318286 -0.474303 -0.314723 0.519197 0.0961656
##  50:     369.67207: 0.254566 0.792503 -0.269716 -0.0708592 0.318285 -0.474301 -0.314721 0.519197 0.0961642
##  51:     369.67207: 0.254566 0.792504 -0.269717 -0.0708597 0.318285 -0.474301 -0.314722 0.519197 0.0961628
##  52:     369.67207: 0.254566 0.792503 -0.269717 -0.0708592 0.318285 -0.474301 -0.314723 0.519197 0.0961629
##  53:     369.67207: 0.254566 0.792504 -0.269718 -0.0708589 0.318285 -0.474302 -0.314723 0.519197 0.0961634
##  54:     369.67207: 0.254566 0.792504 -0.269718 -0.0708588 0.318286 -0.474302 -0.314724 0.519196 0.0961638
##  55:     369.67207: 0.254566 0.792505 -0.269719 -0.0708583 0.318285 -0.474301 -0.314724 0.519196 0.0961637
##  56:     369.67207: 0.254566 0.792505 -0.269719 -0.0708583 0.318285 -0.474301 -0.314724 0.519196 0.0961637
## 
## Final Estimate of the Negative LLH:
##  LLH:  -1609.982    norm LLH:  -4.953792 
##            mu           ar1           ar2           ar3           ar4 
##  5.759787e-04  7.925046e-01 -2.697186e-01 -7.085835e-02  3.182852e-01 
##           ar5           ma1         omega        alpha1 
## -4.743015e-01 -3.147241e-01  2.657932e-06  9.616373e-02 
## 
## R-optimhess Difference Approximated Hessian Matrix:
##                   mu           ar1           ar2           ar3
## mu     -2.403192e+08 -206364.59196 -197930.30572 -191267.10010
## ar1    -2.063646e+05    -922.17191    -674.88218    -352.94345
## ar2    -1.979303e+05    -674.88218    -960.33071    -684.03892
## ar3    -1.912671e+05    -352.94345    -684.03892    -976.46514
## ar4    -1.979589e+05    -204.19124    -393.96725    -721.17024
## ar5    -2.006188e+05    -114.14893    -228.55689    -400.63696
## ma1    -1.012914e+04    -360.44388    -130.41195     -35.28902
## omega  -1.640944e+09 2041476.43332 2577941.39342 1711182.40162
## alpha1  4.053743e+04     -53.07445     -73.52417     -49.37011
##                  ar4           ar5           ma1         omega
## mu     -197958.91678 -200618.80399  -10129.14320 -1.640944e+09
## ar1       -204.19124    -114.14893    -360.44388  2.041476e+06
## ar2       -393.96725    -228.55689    -130.41195  2.577941e+06
## ar3       -721.17024    -400.63696     -35.28902  1.711182e+06
## ar4      -1021.27786    -732.96580     -26.01856 -9.288302e+05
## ar5       -732.96580   -1017.74709     -25.79690 -8.438515e+05
## ma1        -26.01856     -25.79690    -358.45313  1.922555e+06
## omega  -928830.23976 -843851.46618 1922555.34473 -1.954559e+13
## alpha1      20.47444      17.59282     -49.32113 -4.316915e+07
##               alpha1
## mu      4.053743e+04
## ar1    -5.307445e+01
## ar2    -7.352417e+01
## ar3    -4.937011e+01
## ar4     2.047444e+01
## ar5     1.759282e+01
## ma1    -4.932113e+01
## omega  -4.316915e+07
## alpha1 -2.545195e+02
## attr(,"time")
## Time difference of 0.02093601 secs
## 
## --- END OF TRACE ---
## 
## 
## Time to Estimate Parameters:
##  Time difference of 0.123076 secs
## 
## Title:
##  GARCH Modelling 
## 
## Call:
##  garchFit(formula = ~arma(5, 1) + garch(1, 0), data = df22.difflog) 
## 
## Mean and Variance Equation:
##  data ~ arma(5, 1) + garch(1, 0)
## <environment: 0x7fafd74af0a8>
##  [data = df22.difflog]
## 
## Conditional Distribution:
##  norm 
## 
## Coefficient(s):
##          mu          ar1          ar2          ar3          ar4  
##  5.7598e-04   7.9250e-01  -2.6972e-01  -7.0858e-02   3.1829e-01  
##         ar5          ma1        omega       alpha1  
## -4.7430e-01  -3.1472e-01   2.6579e-06   9.6164e-02  
## 
## Std. Errors:
##  based on Hessian 
## 
## Error Analysis:
##          Estimate  Std. Error  t value Pr(>|t|)    
## mu      5.760e-04   8.898e-05    6.473 9.61e-11 ***
## ar1     7.925e-01   7.569e-02   10.470  < 2e-16 ***
## ar2    -2.697e-01   7.112e-02   -3.792 0.000149 ***
## ar3    -7.086e-02   6.768e-02   -1.047 0.295141    
## ar4     3.183e-01   6.479e-02    4.912 9.00e-07 ***
## ar5    -4.743e-01   4.914e-02   -9.652  < 2e-16 ***
## ma1    -3.147e-01   8.139e-02   -3.867 0.000110 ***
## omega   2.658e-06   3.042e-07    8.738  < 2e-16 ***
## alpha1  9.616e-02   8.947e-02    1.075 0.282457    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log Likelihood:
##  1609.982    normalized:  4.953792 
## 
## Description:
##  Thu Apr 30 16:55:59 2020 by user:  
## 
## 
## Standardised Residuals Tests:
##                                 Statistic p-Value     
##  Jarque-Bera Test   R    Chi^2  0.2734848 0.8721949   
##  Shapiro-Wilk Test  R    W      0.9962619 0.6439345   
##  Ljung-Box Test     R    Q(10)  30.02556  0.0008484318
##  Ljung-Box Test     R    Q(15)  45.6876   5.960481e-05
##  Ljung-Box Test     R    Q(20)  59.62209  8.147078e-06
##  Ljung-Box Test     R^2  Q(10)  16.71716  0.08086142  
##  Ljung-Box Test     R^2  Q(15)  19.28223  0.2012282   
##  Ljung-Box Test     R^2  Q(20)  36.25517  0.0143522   
##  LM Arch Test       R    TR^2   16.92929  0.1522767   
## 
## Information Criterion Statistics:
##       AIC       BIC       SIC      HQIC 
## -9.852199 -9.747417 -9.853679 -9.810380

garch_uk1=garch(residual_arima511,order=c(1,1),trace=F)

## Warning in garch(residual_arima511, order = c(1, 1), trace = F): singular
## information

summary(garch_uk1)

## 
## Call:
## garch(x = residual_arima511, order = c(1, 1), trace = F)
## 
## Model:
## GARCH(1,1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8507 -0.3906  0.2907  0.9232  2.9349 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 2.383e+09          NA       NA       NA
## a1 2.094e-02          NA       NA       NA
## b1 1.493e-01          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 0.13246, df = 2, p-value = 0.9359
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.34731, df = 1, p-value = 0.5556

AIC(garch_uk1) ## AIC is smaller

## [1] 8003.569

garch_uk2=garch(residual_arima511,order=c(0,1),trace=F)

## Warning in garch(residual_arima511, order = c(0, 1), trace = F): singular
## information

summary(garch_uk2)

## 
## Call:
## garch(x = residual_arima511, order = c(0, 1), trace = F)
## 
## Model:
## GARCH(0,1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.0053 -0.4058  0.3059  0.9611  3.1050 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 2.515e+09          NA       NA       NA
## a1 3.916e-02          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 0.089604, df = 2, p-value = 0.9562
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.94383, df = 1, p-value = 0.3313

AIC(garch_uk2)

## [1] 8003.624

Canada

df1_ca=subset(ifs, (Indicator_Code=='LE_PE_NUM') & (Country_Name=='Canada'))
df1_ca$Time_Period <- gsub("M", "/", df1_ca$Time_Period)

df1_ca['date'] = '/01'
df1_ca$Time_Period <- paste(df1_ca$Time_Period, df1_ca$date,  sep="")

df1_ca$Time_Period = as.Date(df1_ca$Time_Period, "%Y/%m/%d")

df1_ca = df1_ca[order(df1_ca$Time_Period),]
df1_ca = subset(df1_ca, select = -c(date))
rownames(df1_ca)=NULL
df1_ca=na.omit(df1_ca)

df33=ts(df1_ca$Value, frequency = 12, start = c(1993, 1))
plot(df33, main="Canada: Employment, Persons, Number of")

1.Decompose Analysis

plot(decompose(df33))

## There is a increasing trend with little drop in 2018
## There is a obvious seasonality

2.Spectral Analysis

sp=spectrum(df33, log='no')

# find the cycle with predominant frequecny 
sp$fre[which.max(sp$spec)] #  0.0375

## [1] 0.0375

# find the cycle of predominant frequecny 
1/sp$fre[which.max(sp$spec)] # 26.7 month

## [1] 26.66667

max(sp$spec) #estimate of spectral density at predominant period which is close to 503508181908.

## [1] 503508181908

U = qchisq(.025,2)
L = qchisq(.975,2)

2*max(sp$spec)/L #lower bound for spectral density

## [1] 136493530941

2*max(sp$spec)/U #upper bound for spectral density

## [1] 1.988751e+13

# The 95% confidence interval is [132631376982, 1.932478e+13]

3. ARIMA model

df33.diff = diff(df33)
plot(df33.diff)

acf2(df33.diff)#p = 1, q = 4,5

##         ACF  PACF
##  [1,]  0.37  0.37
##  [2,]  0.14  0.00
##  [3,] -0.13 -0.21
##  [4,] -0.32 -0.25
##  [5,] -0.36 -0.18
##  [6,] -0.27 -0.09
##  [7,] -0.37 -0.39
##  [8,] -0.34 -0.43
##  [9,] -0.13 -0.31
## [10,]  0.14 -0.21
## [11,]  0.36 -0.22
## [12,]  0.92  0.82
## [13,]  0.35 -0.24
## [14,]  0.13 -0.09
## [15,] -0.13 -0.02
## [16,] -0.31 -0.07
## [17,] -0.35 -0.02
## [18,] -0.26  0.01
## [19,] -0.36 -0.01
## [20,] -0.32  0.09
## [21,] -0.12 -0.10
## [22,]  0.14 -0.02
## [23,]  0.34 -0.05
## [24,]  0.88  0.13
## [25,]  0.33 -0.09
## [26,]  0.12 -0.02
## [27,] -0.13  0.01
## [28,] -0.30 -0.04
## [29,] -0.34 -0.06
## [30,] -0.24  0.05
## [31,] -0.35 -0.01
## [32,] -0.31 -0.06
## [33,] -0.12 -0.02
## [34,]  0.14 -0.04
## [35,]  0.34  0.07
## [36,]  0.84  0.02
## [37,]  0.32 -0.02
## [38,]  0.11 -0.03
## [39,] -0.12  0.02
## [40,] -0.28  0.02
## [41,] -0.33  0.00
## [42,] -0.23 -0.01
## [43,] -0.34  0.01
## [44,] -0.31 -0.01
## [45,] -0.11 -0.02
## [46,]  0.14 -0.02
## [47,]  0.33 -0.03
## [48,]  0.80  0.05

arima411 = Arima(df33,order=c(4,1,1),include.drift = T)
arima511 = Arima(df33,order=c(5,1,1),include.drift = T)
AIC(arima411)

## [1] 8307.449

AIC(arima511)

## [1] 8304.877

BIC(arima411)

## [1] 8333.695

BIC(arima511)

## [1] 8334.872

auto.arima(df33,seasonal = F)

## Series: df33 
## ARIMA(4,1,1) with drift 
## 
## Coefficients:
##         ar1      ar2      ar3      ar4      ma1      drift
##       0.956  -0.1363  -0.1455  -0.1848  -0.8778  19618.209
## s.e.  0.060   0.0775   0.0774   0.0584   0.0269   1818.469
## 
## sigma^2 estimated as 1.753e+10:  log likelihood=-4146.72
## AIC=8307.45   AICc=8307.82   BIC=8333.69

#arima511 has lower AIC and BIC

## SEASONA ARIMA
fit33 <- auto.arima(df33, trace = TRUE) #  Best model: ARIMA(0,1,0)(1,1,2)[12]

## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)(1,1,1)[12]                    : 7018.275
##  ARIMA(0,1,0)(0,1,0)[12]                    : 7106.15
##  ARIMA(1,1,0)(1,1,0)[12]                    : 7044.461
##  ARIMA(0,1,1)(0,1,1)[12]                    : 7009.45
##  ARIMA(0,1,1)(0,1,0)[12]                    : 7107.929
##  ARIMA(0,1,1)(1,1,1)[12]                    : 7019.621
##  ARIMA(0,1,1)(0,1,2)[12]                    : 7011.272
##  ARIMA(0,1,1)(1,1,0)[12]                    : 7045.88
##  ARIMA(0,1,1)(1,1,2)[12]                    : 7010.959
##  ARIMA(0,1,0)(0,1,1)[12]                    : 7007.426
##  ARIMA(0,1,0)(1,1,1)[12]                    : 7017.631
##  ARIMA(0,1,0)(0,1,2)[12]                    : 7009.225
##  ARIMA(0,1,0)(1,1,0)[12]                    : 7044.131
##  ARIMA(0,1,0)(1,1,2)[12]                    : 7008.907
##  ARIMA(1,1,0)(0,1,1)[12]                    : 7011.53
##  ARIMA(1,1,1)(0,1,1)[12]                    : 7010.49
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(0,1,0)(0,1,1)[12]                    : 7289.008
## 
##  Best model: ARIMA(0,1,0)(0,1,1)[12]

sarima(df33,0,1,0,1,1,2,12)

## initial  value 10.811485 
## iter   2 value 10.689700
## iter   3 value 10.648455
## iter   4 value 10.644581
## iter   5 value 10.642988
## iter   6 value 10.642576
## iter   7 value 10.640678
## iter   8 value 10.638508
## iter   9 value 10.637715
## iter  10 value 10.637566
## iter  11 value 10.637474
## iter  12 value 10.637417
## iter  13 value 10.637138
## iter  14 value 10.637029
## iter  15 value 10.636995
## iter  15 value 10.636995
## final  value 10.636995 
## converged
## initial  value 10.642761 
## iter   2 value 10.642536
## iter   3 value 10.642266
## iter   4 value 10.642265
## iter   4 value 10.642265
## iter   4 value 10.642265
## final  value 10.642265 
## converged

## Warning in sqrt(diag(fitit$var.coef)): NaNs produced

## Warning in sqrt(diag(fitit$var.coef)): NaNs produced

## $fit
## 
## Call:
## stats::arima(x = xdata, order = c(p, d, q), seasonal = list(order = c(P, D, 
##     Q), period = S), include.mean = !no.constant, transform.pars = trans, fixed = fixed, 
##     optim.control = list(trace = trc, REPORT = 1, reltol = tol))
## 
## Coefficients:

## Warning in sqrt(diag(x$var.coef)): NaNs produced

##          sar1   sma1     sma2
##       -0.6881  0.013  -0.4662
## s.e.      NaN    NaN      NaN
## 
## sigma^2 estimated as 1.711e+09:  log likelihood = -3642.48,  aic = 7292.97
## 
## $degrees_of_freedom
## [1] 299
## 
## $ttable
##      Estimate  SE t.value p.value
## sar1  -0.6881 NaN     NaN     NaN
## sma1   0.0130 NaN     NaN     NaN
## sma2  -0.4662 NaN     NaN     NaN
## 
## $AIC
## [1] 23.30021
## 
## $AICc
## [1] 23.30046
## 
## $BIC
## [1] 23.34763

predict

pred33=forecast(fit33,h=48)
pred33

##          Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## Apr 2019       18769843 18716639 18823047 18688474 18851212
## May 2019       19174437 19099195 19249679 19059364 19289510
## Jun 2019       19365885 19273733 19458038 19224950 19506821
## Jul 2019       19299169 19192760 19405577 19136431 19461907
## Aug 2019       19259613 19140644 19378581 19077666 19441559
## Sep 2019       19137818 19007495 19268141 18938506 19337130
## Oct 2019       19161719 19020954 19302485 18946437 19377001
## Nov 2019       19145259 18994775 19295744 18915113 19375405
## Dec 2019       19094805 18935192 19254418 18850698 19338912
## Jan 2020       18856984 18688737 19025231 18599673 19114295
## Feb 2020       18916639 18740180 19093098 18646769 19186509
## Mar 2020       18935045 18750740 19119350 18653175 19216915
## Apr 2020       19040188 18842877 19237499 18738427 19341949
## May 2020       19444782 19235271 19654293 19124362 19765201
## Jun 2020       19636230 19415191 19857269 19298181 19974280
## Jul 2020       19569514 19337519 19801508 19214709 19924319
## Aug 2020       19529958 19287502 19772413 19159154 19900762
## Sep 2020       19408163 19155679 19660647 19022022 19794303
## Oct 2020       19432064 19169936 19694193 19031173 19832955
## Nov 2020       19415604 19144173 19687035 19000487 19830721
## Dec 2020       19365150 19084725 19645574 18936278 19794022
## Jan 2021       19127329 18838190 19416468 18685129 19569528
## Feb 2021       19186984 18889386 19484582 18731847 19642120
## Mar 2021       19205390 18899567 19511213 18737674 19673106
## Apr 2021       19310533 18992384 19628681 18823967 19797099
## May 2021       19715127 19385113 20045140 19210414 20219839
## Jun 2021       19906575 19565108 20248043 19384346 20428804
## Jul 2021       19839858 19487310 20192407 19300681 20379035
## Aug 2021       19800303 19437010 20163595 19244695 20355911
## Sep 2021       19678508 19304780 20052235 19106941 20250075
## Oct 2021       19702409 19318530 20086288 19115317 20289501
## Nov 2021       19685949 19292180 20079717 19083732 20288166
## Dec 2021       19635495 19232079 20038911 19018523 20252466
## Jan 2022       19397674 18984836 19810512 18766293 20029055
## Feb 2022       19457329 19035279 19879378 18811859 20102798
## Mar 2022       19475735 19044670 19906799 18816478 20134991
## Apr 2022       19580878 19137223 20024533 18902366 20259389
## May 2022       19985471 19529574 20441369 19288236 20682707
## Jun 2022       20176920 19709100 20644740 19461451 20892390
## Jul 2022       20110203 19630757 20589650 19376953 20843453
## Aug 2022       20070647 19579850 20561445 19320038 20821257
## Sep 2022       19948853 19446961 20450744 19181276 20716429
## Oct 2022       19972754 19460008 20485500 19188577 20756931
## Nov 2022       19956294 19432919 20479669 19155861 20756726
## Dec 2022       19905839 19372047 20439632 19089474 20722205
## Jan 2023       19668019 19124008 20212029 18836026 20500011
## Feb 2023       19727674 19173633 20281714 18880342 20575005
## Mar 2023       19746079 19182188 20309971 18883682 20608477

autoplot(pred33)

## 4. ARCH/GARCH

fit4=auto.arima(df33,  seasonal = FALSE, trace = TRUE) # ARIMA(5,1,1)

## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,1,2)            with drift         : 8328.876
##  ARIMA(0,1,0)            with drift         : 8423.307
##  ARIMA(1,1,0)            with drift         : 8379.721
##  ARIMA(0,1,1)            with drift         : 8387.116
##  ARIMA(0,1,0)                               : 8425.806
##  ARIMA(1,1,2)            with drift         : 8377.805
##  ARIMA(2,1,1)            with drift         : Inf
##  ARIMA(3,1,2)            with drift         : 8238.029
##  ARIMA(3,1,1)            with drift         : 8295.752
##  ARIMA(4,1,2)            with drift         : 8287.616
##  ARIMA(3,1,3)            with drift         : 8328.485
##  ARIMA(2,1,3)            with drift         : 8329.249
##  ARIMA(4,1,1)            with drift         : 8285.513
##  ARIMA(4,1,3)            with drift         : Inf
##  ARIMA(3,1,2)                               : Inf
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(3,1,2)            with drift         : Inf
##  ARIMA(4,1,1)            with drift         : 8307.815
## 
##  Best model: ARIMA(4,1,1)            with drift

df33.diff=diff(df33)
plot(df33.diff, main=" diff data")

df33.log=log(df33)
plot(df33.log, main=" log data")

df33.difflog=diff(log(df33))
plot(df33.difflog, main=" difflog data")

acf(df33.difflog, lag.max = 100) # from pacf plot, we could observe the seasonality

pacf(df33.difflog, lag.max = 100)# from pacf plot, AR(2) or AR(4)

acf2(df33^2)

##        ACF  PACF
##  [1,] 0.99  0.99
##  [2,] 0.97 -0.12
##  [3,] 0.95 -0.03
##  [4,] 0.93  0.02
##  [5,] 0.92  0.11
##  [6,] 0.91  0.11
##  [7,] 0.90  0.05
##  [8,] 0.90  0.08
##  [9,] 0.89  0.07
## [10,] 0.89  0.05
## [11,] 0.89  0.00
## [12,] 0.88 -0.05
## [13,] 0.87 -0.28
## [14,] 0.85 -0.03
## [15,] 0.84 -0.01
## [16,] 0.82 -0.01
## [17,] 0.81  0.07
## [18,] 0.80  0.04
## [19,] 0.79  0.02
## [20,] 0.78  0.03
## [21,] 0.78  0.05
## [22,] 0.78  0.03
## [23,] 0.78  0.01
## [24,] 0.77  0.00
## [25,] 0.76 -0.19
## [26,] 0.74 -0.02
## [27,] 0.73  0.00
## [28,] 0.71  0.01
## [29,] 0.70  0.02
## [30,] 0.69  0.00
## [31,] 0.68  0.01
## [32,] 0.68  0.03
## [33,] 0.67  0.04
## [34,] 0.67  0.02
## [35,] 0.67 -0.02
## [36,] 0.66  0.00
## [37,] 0.65 -0.15
## [38,] 0.64  0.00
## [39,] 0.62  0.00
## [40,] 0.61  0.01
## [41,] 0.59  0.00
## [42,] 0.58 -0.01
## [43,] 0.58  0.00
## [44,] 0.57  0.01
## [45,] 0.57  0.02
## [46,] 0.56  0.01
## [47,] 0.56  0.00
## [48,] 0.56 -0.01

arima511.2=arima(df33, order=c(5,1,1))
residual_arima511.2 = arima511.2$residuals
acf2(residual_arima511.2^2)

##         ACF  PACF
##  [1,] -0.19 -0.19
##  [2,] -0.30 -0.35
##  [3,] -0.05 -0.23
##  [4,]  0.20  0.03
##  [5,]  0.00 -0.01
##  [6,] -0.22 -0.18
##  [7,]  0.00 -0.09
##  [8,]  0.26  0.14
##  [9,] -0.05  0.00
## [10,] -0.28 -0.19
## [11,] -0.18 -0.38
## [12,]  0.78  0.66
## [13,] -0.19 -0.06
## [14,] -0.29 -0.05
## [15,] -0.03 -0.03
## [16,]  0.21  0.06
## [17,]  0.02  0.03
## [18,] -0.22 -0.01
## [19,] -0.01 -0.03
## [20,]  0.23 -0.16
## [21,] -0.03  0.03
## [22,] -0.27 -0.05
## [23,] -0.16 -0.06
## [24,]  0.70  0.14
## [25,] -0.16  0.03
## [26,] -0.26  0.05
## [27,] -0.04 -0.03
## [28,]  0.23  0.12
## [29,]  0.00 -0.10
## [30,] -0.20  0.04
## [31,] -0.01  0.02
## [32,]  0.25  0.07
## [33,] -0.03 -0.05
## [34,] -0.26 -0.03
## [35,] -0.14  0.03
## [36,]  0.65  0.11
## [37,] -0.15  0.06
## [38,] -0.24  0.01
## [39,] -0.02  0.07
## [40,]  0.21 -0.04
## [41,] -0.01  0.04
## [42,] -0.19  0.02
## [43,] -0.02 -0.03
## [44,]  0.21 -0.13
## [45,] -0.03 -0.03
## [46,] -0.26 -0.02
## [47,] -0.14 -0.10
## [48,]  0.62  0.12

garch_ca1=garch(residual_arima511.2,order=c(4,1),trace=F)

## Warning in garch(residual_arima511.2, order = c(4, 1), trace = F): singular
## information

summary(garch_ca1)

## 
## Call:
## garch(x = residual_arima511.2, order = c(4, 1), trace = F)
## 
## Model:
## GARCH(4,1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.7583 -0.1879  0.2742  0.8254  3.0769 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 1.350e+10          NA       NA       NA
## a1 7.244e-05          NA       NA       NA
## b1 6.394e-02          NA       NA       NA
## b2 6.652e-02          NA       NA       NA
## b3 6.925e-02          NA       NA       NA
## b4 6.698e-02          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 1.2106, df = 2, p-value = 0.5459
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 11.595, df = 1, p-value = 0.0006614

AIC(garch_ca1) ## AIC is smaller

## [1] 8266.518

garch_ca2=garch(residual_arima511.2,order=c(0,1),trace=F) # (q,p) # AIC is smaller

## Warning in garch(residual_arima511.2, order = c(0, 1), trace = F): singular
## information

summary(garch_ca2)

## 
## Call:
## garch(x = residual_arima511.2, order = c(0, 1), trace = F)
## 
## Model:
## GARCH(0,1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.8620 -0.1929  0.2840  0.8508  3.1926 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)
## a0 1.710e+10          NA       NA       NA
## a1 2.048e-04          NA       NA       NA
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 1.439, df = 2, p-value = 0.487
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 11.979, df = 1, p-value = 0.0005381

AIC(garch_ca2)

## [1] 8337.543

par(mfrow = c(1,3))
plot(df11,ylab = "Number", main="United States: Employment")
plot(df22,ylab = "Number", main="United Kingdom: Employment")
plot(df33,ylab = "Number", main="Canada: Employment")

Employment, Persons, Number of

Ning Hu

4/2/2020

Research Indicator: Employment, Persons, Number of

Indicator Code: LE_PE_NUM

United States

2.Spectral Analysis

3.ARIMA model

Prediction

4.ARCH/GARCH model

United Kingdom

1. decompose Analysis

2. Spectral Analysis

3. ARIMA model

Prediction

4. ARCH/GARCH model

Canada

1.Decompose Analysis

2.Spectral Analysis

3. ARIMA model

predict